Factor Friend or Foe?

Explicit Instruction/Teacher modeling

• Introduce the vocabulary words to the students by displaying the vocabulary cards, one at a time, on the document camera. Read aloud the word, its definition, and describe the image if available.
• Ask students to share additional information or examples they have on the vocabulary terms and add them to the vocabulary cards. For example, if students know the synonym in their home language (L1), add it to card.
• Focus on the concept of factors. Explain that it is important to be able to find all the factors of a product because it helps us understand various ways to make the number and it also helps us connect multiplication with division, which happens to be the inverse, or opposite, operation.
• Show students how to find the factors of the number 8 with the factor rainbow strategy. Draw an arch with 1 at the end of the arch and 8 at the other end. Explain that one and the number itself are always factors of the given number. The next step is to draw a smaller arch under the first arch and ask yourself, what other two numbers can be multiplied to get the product of 8? Tell students that it is helpful to try dividing the product by small numbers such as 2, 3, and 4 to see if they are factors. Since multiplication and division are inverse operations, we can use both operations to help us find the factors. In this case, 2 x 4 = 8 so write 2 and 4 at each end of the arch. We can also think of it as 8 ÷ 2 = 4. We now know there are no other factors of 8 since 3 did not work. This means that the factors of 8 are 1, 2, 4, and 8.
• Model rainbow factorization with another number such as 12 or 16.

Guided Practice

• Explain to students that finding the factors essentially means that we are separating the product evenly. For example, say, "If I had 15 cookies, I wonder how I could break them up into equal groups. I realize that the factors 1 and 15 mean one person could have all 15 cookies or 15 people could each have one cookie. The other factor pair is 3 and 5 which means that 3 people could each get 5 cookies or 5 people could get 3 cookies."
• Place students in partnerships and give them scratch paper or their math journals. Have them try the rainbow factor strategy with simple numbers such as 10, 15, or 18.
• Have students volunteer to share their math work on the document camera. Provide the following sentence starters to help students share.
  • "First, we drew an arch and wrote..."
  • "Then, we divided ____ (product) by 2/3/4 and discovered that ____ and ____ are also factors."
  • "After that, we tried..."
  • "All the factors of ____ (product) are..."
• Listen in on students' responses and correct any errors in their math thinking.

Group work time

• Ask students to consider how they might find out if a number is not a factor of a product. Sample answers include: "If we can't multiply the number by another number to get the product," or "If we divide the product by the number and we can't get a whole number as an answer, then we know it's not a factor."
• Distribute the Factor or Not? worksheet to students and display a copy on the document camera. Read aloud the directions and explain the structure of the table provided. Tell students that they will work with a partner to complete the table. Explain that the paragraph frame is given to provide them language support to explain their math thinking process.
• Hand out scratch paper to each student and tell them to use the rainbow factor strategy to help them determine which number does not belong in the factor list.
• Call on a student to help you model how to work together and discuss the answer. Tell students that both partners should look at the second row in the table and consider the information. For example, say "The product is 15. The table lists the factors as 1, 3, 5, 7, and 15. I will do the factor rainbow strategy on my scratch paper. I discover that the factors are 1, 3, 5, and 15. So I realize that 7 is not a factor because we can't multiply anything by 7 to get 15. Do you agree?" Give the students a chance to agree or disagree before completing the paragraph frame together.
• Allow ample time for pairs to complete the worksheet together. Encourage students to talk through the problems and any disagreements that arise.
• Have some students share their responses with the whole group. Make sure they provide their reasoning in complete sentences.